Select one manipulative and describe how it could enhance the understanding of a mathematical concept or big idea. Align the tool to a grade and suggest a hands-on task.

Tangram is a puzzle made from 7 differently-sized shapes that, when placed in a specific order, create an image. Tangrams are a great way to reinforce themes within geometry, such as shapes, sides, angles, and vertices. Additionally, through the continued use of tangrams, students are encouraged to recognize shapes and mathematical themes in their everyday life. The different sizes, shapes, and colours add to the sensory stimulation that engages students and keeps them attended to the activity at hand.

I have seen great success in using tangrams in grades as early as kindergarten. Students can be prompted to create an object using tangrams, contributing to their spatial awareness, creativity, and problem solving. Students can also be provided with silhouettes in which they must make their tangram pieces fit. This type of activity has now be enhanced through the use of the Osmo tangram application. The Osmo uses an iPad, tangram shapes, and a mirrored lens over the iPad’s camera to create a digital game that has students align their shapes on the table into the shape displayed on the iPad. The application then takes them through levels and different tangram puzzles using the same shapes. Here’s some more information about this engaging task:

Post 3 of the most important strategies or things a teacher is doing in the math classroom to support and/or create an inviting math environment. Describe and reflect on these high yield strategies.

Educators across the system are working towards creating positive, safe, and supportive learning environments in every subject matter. The Guide to Effective Instruction in Mathematics discusses the importance of developing a mathematical community, in which students can learn to support each other in their learning and build off of each other’s thinking. This is made possible when using strategies like Bansho or Number Talks. These strategies allow student to share their thought processes, while also learning from the strategies being shared by their peers. Intentionally teaching students how to be positive and respectful members of a learning community helps to add to the overall learning experience.

The using classroom resources is especially important with our new wave of teaching mathematics. Even a short 10 years ago, I was being taught mathematics in a learning environment that only used resources such as a textbook and worksheets. Nowadays, we provide experiential learning opportunities for our students in which they can touch, feel, and move throughout their learning. This involves using resources like manipulatives and technology (laptop, tablets, tools, etc.) to keep the learning engaging, while also teaching the students important 21st Century learning skills.

Structure in the form of the three-part math lesson is an important strategy when creating an inviting math environment. The area in this type of lesson that is the first to be disregarded is the consolidation, typically because the class runs out of time. By being more structured with the timing of specific types and sections of lessons, educators can ensure that the consolidation is carried out for all lessons. This allows students to share their learning, use appropriate vocabulary, discuss strategies, and learn from their peers.

Feedback is often secondary to the grade or level received because it has been ingrained in the world for so long that the final grade is what counts. Educational research generally says that feedback without a mark is the most powerful for affecting change.

How does this research impact your assessment practices? How do you use feedback to move student thinking forward?

Even as a new teacher entering into the field of education, I have met a number of teachers who are going gradeless in their classrooms. This aligns with the educational research and their own experiences which tell them that students respond best to written or verbal feedback, rather than a letter or percentage grade. Too often, students look at the final grade and take it as the ‘be all and end all’, skipping over the descriptive feedback provided about their current performance and ways to improve. This cycle also leads students to “only want a C” or to achieve the letter grade that meets their parent’s expectations. This, however, actually takes away from the learning process in that final grades are the smallest form of feedback for students; it labels their current ability without providing ways or suggestions for improving.

The Assessment and Evaluation of Student Achievement portion of the Ontario mathematics curriculum states: “As part of assessment, teachers provide students with descriptive feedback that guides their efforts towards improvement” (Ontario Mathematics Curriculum, 2005). Education involves learning, trying, receiving feedback, and trying again. Written and/or oral feedback provides students with a personalized description of how to improve. This is why more and more teachers are moving away from overall grades and focusing on detailed feedback. However, I also acknowledge that much of our education system past the elementary grades revolves around percentages and final outcomes, especially because that determines the future of a student’s education (i.e., next course, university acceptance, etc.). I believe that it is our duty as Primary/Junior teachers to provide students with the understanding that feedback is important and that there is still something to learn when receiving feedback (learning about our process, as well as during the process).

Share your thoughts on whether you agree or disagree with Marion Small’s view of success criteria and “generalizing vs particularizing” in math.

Marion Small makes a very interesting point when she discusses generalizing vs. particularizing in her video about success criteria. Marion spoke about how some educators identify the success of their students when they use the mathematic terminology that the researchers and textbooks provide. In these cases, there is a higher emphasis on ‘particularizing’, in that the students must learn to remember the name of a strategy rather than being able to explain how a strategy was used. ‘Generalizing’ involves having the students explain what method they used and how it was effective, without hyper-focusing on terminology.

I agree with Marion Small’s view of generalizing and particularizing in math, especially as it relates to success criteria. In my own practicum placements, I facilitated math number talks and explored many strategies to solve the same equation. Personally, I found it difficult to remember the various names given to each of the strategies used (i.e., double plus one, decomposing numbers, friendly numbers, etc.). I came to the realization that if it was difficult for me to remember the terminology, it was probably difficult for my students. Additionally, I had to consider what my true success criteria were for my students during the number talks: Was it to use the proper name of the strategy used, or to utilize multiple strategies and be able to explain what they did? For my group of students, the function was more important than the lingo, and I believe my students learned more from sharing strategies with their peers than from putting names to strategies. This does not dismiss the importance of the language used in math classes, but it shifts the focus from particularizing to generalizing.

What do you think will have the biggest impact on student engagement, motivation, and success in the elementary mathematics classroom?

In my opinion, the delivery of the mathematic content will have the largest impact on student engagement, motivation, and success. The best way that we can engage our students is by making the learning fun and interesting. We have a set curriculum that we must follow, so there isn’t much that we can change about what is being taught, but we can alter how it is being delivered. By using more play-based learning throughout the elementary grades, students will find consistency in how they learn mathematics while also associating a typically “difficult” subject with fun learning. It is also important that educators use visually stimulating tools to engage and motivate their students throughout their learning, including videos, books, and other texts (What Works? Research into Practice, pg. 2-3). When we are able to grasp our students attention and engage them in fun learning, students will be both intrinsically and extrinsically motivated to achieve success.

To ensure the success of all students in our elementary mathematics classrooms, we must ensure that the content being delivered is at an grade-appropriate and obtainable level. The lessons should be differentiated to meet the levels of all students in the classroom, as well as providing multiple entry points into the same learning opportunities. One thing that the Differentiating Mathematics Instruction Capacity Building Series suggests considering when differentiating our instruction are the students’ Zones of Proximal Development. This outlines the “distance between the actual developmental level” of the student and their “level of potential development” (Capacity Building Series, pg. 1). When we take this into account, we can determine where they are currently with their learning and provide them with learning opportunities that can challenge and extend this learning.

Reflect on the value of problem solving and consider what makes a rich and engaging question. Discuss how important it is for students to explore and use communication in consolidating mathematical understanding through ‘math talk’.

Problem solving questions, specifically in math, provide opportunities for students to practice their learned skills in applicable and relevant situations. They challenge the students to reflect on what they have learned theoretically and apply this knowledge in practical, thought-provoking applications. It is very important that we teach students to embrace problem solving, treating it like a puzzle to be solved rather than a brick wall preventing us from achieving success. When we adequately prepare students with the tools that they need during problem solving, they come to learn that they are able to problem solve and they can achieve success. This, in turn, develops a positive disposition towards problem solving for our students.

The Guide to Effective Instruction: Grades K to 6 – Volume 2 – Problem Solving and Communication teaches us that rich and engaging problem solving questions not only teach students through problem solving (practicing conceptual understanding), but they also teach student about problem solving by learning applicable learning skills (Guide to Effective Instruction, pg. 6). By teaching student through and about problem solving, we are able to see if the student has grasped the concept while also exploring the strategy they used throughout the process. When we are able to see both aspects, we then know that we have created a rich mathematical question. It is also important that we ensure the questions are relevant to the students by using real-world situations that are linked to their specific interests.

Conversations around problem solving help to teach students to be cognitive about their own strategies, while also being able to learn from their peers and adopt new and perhaps more efficient strategies. As Marion Small says in the video Open Questions and Contexts, “[Different strategies] enrich the conversation; it does not detract from it.” Math talks and bansho consolidation presentations are great ways to verbally explore these strategies in a whole-class setting. Other ways to communicate their thinking could be in a math journal, in which the student explains the strategies they used throughout the day’s lesson, or by creating a video/voice recording of their verbal explanations (for those students less inclined to share with the class).

After spending time researching and exploring different teaching models, choose one to summarize, providing suggestions on how to incorporate this approach into your math class. Post your model along with a brief description of it and its application and usefulness in the primary/junior classroom.

Teaching Model

Doug Clements: Intentional Play-based Learning

Brief Description

Educators must stop choosing either a strictly “play-based” or strictly “academic” approach to teaching/learning mathematics

Extreme play-based approaches to learning, where the teacher is completely removed from the learning and the students are in full control of their play, is not the essence of the best play-based curriculum

Extreme academic approaches, where the students sit at desks and answer problem after problem, produce mechanical, uncreative thinkers

The best type of learning including all kinds of learning experiences, including both play-based and guided learning

Educators should prompt students and give creative challenges that develop high-caliber mathematical thinking and reasoning while the students are engaging in play-based learning

Application and Usefulness

Kids develop higher levels of social skills, emotional skills, and self-regulation skills when they emerge in guided play-based learning

Learning is enhanced when students can plan and established roles during their play, while in a guided environment

Need to talk about the mathematic sand development the appropriate language to convey learning, which can be further assisted by an educator guiding the engaging in the student’s play

Suggestions for Integration

Play-based learning should be purposeful with some pre-determined structure

Challenges could be presented throughout the play-based learning to encourage further extensions of learning and tier the expectations for specific groupings or individuals

Check-ins with students throughout the learning helps to reinforce mathematical language and develops the student’s ability to explain their processes and strategies

Share a minimum of two connections that integrate math with other subjects. Post these two ideas along with an explanation as to why cross-curricular connections are beneficial to student learning, particularly as it relates to learning mathematics.

Coding is a great cross-curricular connection between Math, Science, and even Language. Coding has students create sequences of commands that lead to a specific action or outcome. When using robotics technology, such as a Sphero, students are able to code the robot to move a certain distance, rotate a specific way, and even travel at a certain speed. Robotics and coding would typically fall under the category of Science and Technology, however, there are many different and creative ways in which it can have a math focus. For example, students could use angles and rotations to maneuver the robot through a maze that the students create. There are also some valuable Language expectations met when coding, predominantly procedural writing. Coding and robotics are great ways to bring a math problem to life, while also teaching the students valuable and applicable 21st century skills.

There are many ways in which Math can also be cross-curricular with Geography. On way in particular that I was able to make a cross-curricular connection between these two subjects was when our class was learning about the environment and natural resources. Students used their data management skills to create and conduct a survey to other students within the school about the amount of waste that they brought in their lunches each day. The students were then able to use this data to calculate how much waste the school would produce each week, month, and school year, while also using different weight measurements. This proved to not only hit a number of different curriculum expectations in math, but it also helped the students to grasp the severity of their waste production from a geography mindset.